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A002763 Number of bipartite partitions.

%I A002763 M3414 N1380 #33 Dec 20 2021 20:18:42
%S A002763 4,11,26,52,98,171,289,467,737,1131,1704,2515,3661,5246,7430,10396,
%T A002763 14405,19760,26884,36269,48583,64614,85399,112170,146526,190362,
%U A002763 246099,316621,405556,517224,657012,831320,1048055,1316611,1648486,2057324,2559719,3175309
%N A002763 Number of bipartite partitions.
%D A002763 M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
%D A002763 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002763 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002763 Alois P. Heinz, <a href="/A002763/b002763.txt">Table of n, a(n) for n = 0..300</a>
%H A002763 M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)
%F A002763 a(n) = a(n-1) + A000041(n) + A000070(n) + A000291(n), for n>0 - _Alford Arnold_, Dec 10 2007
%F A002763 From _Vaclav Kotesovec_, Jan 07 2017: (Start)
%F A002763 G.f.: (4 - x - 3*x^2 + x^3) / ((1-x)^3 * (1+x)) * Product_{k>=1} 1/(1-x^k).
%F A002763 a(n) ~ exp(Pi*sqrt(2*n/3)) * 3*sqrt(n)/(2*sqrt(2)*Pi^3).
%F A002763 (End)
%p A002763 with(numtheory):
%p A002763 b:= proc(n, k) option remember;
%p A002763       `if`(n>k, 0, 1) +`if`(isprime(n), 0,
%p A002763       add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
%p A002763     end:
%p A002763 a:= n-> b((45*2^n)$2):
%p A002763 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 26 2013
%t A002763 b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[45*2^n, 45*2^n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Mar 20 2014, after _Alois P. Heinz_ *)
%t A002763 nmax = 100; CoefficientList[Series[(4 - x - 3*x^2 + x^3) / ((1 - x)^3 * (1 + x)) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 07 2017 *)
%Y A002763 Cf. A082775, A129306.
%K A002763 nonn
%O A002763 0,1
%A A002763 _N. J. A. Sloane_
%E A002763 Extended beyond a(25) by _Alois P. Heinz_, May 26 2013